×

zbMATH — the first resource for mathematics

Lyapunov stability of ground states of nonlinear dispersive evolution equations. (English) Zbl 0594.35005
The orbital stability of the ground state solutions of the nonlinear Schrödinger equation \[ (1)\quad i\phi_ t(x,t)+\Delta \phi (x,t)+f(| \phi (x,t)|^ 2)\phi (x,t)=0,\quad x\in {\mathbb{R}}^ N \] and the generalized Korteweg-de Vries equation \[ (2)\quad w_ t(x,t)+a(w(x,t))w_ x(x,t)+w_{xxx}(x,t)=0,\quad x\in {\mathbb{R}} \] are studied using a Lyapunov function which is a conserved energy integral. In the case of (1) the ground state is orbitally stable, in the particular case where \(f(| \phi |^ 2)=| \phi |^{2\sigma}\), if \(\sigma <2/N\) and \(N=1\) or \(N=3\). A more general class of functions f is also considered. In the case of (2), if R(E) is a ground state with energy E, then a solitary wave \(\psi(x-ct)\) is orbitally stable if \(\phi(c)=(d/dc)\| R(c)\|^ 2>0.\)
Reviewer: S.P.Banks

MSC:
35B35 Stability in context of PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35J10 Schrödinger operator, Schrödinger equation
58J47 Propagation of singularities; initial value problems on manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Akhamanov, Sov. Phys. JETP 23 pp 1025– (1966)
[2] Arnold, Am. Math. Soc. Transl. 79 pp 267– (1969) · Zbl 0191.56303
[3] Benjamin, Proc. R. Soc. Lond. A 328 pp 153– (1972)
[4] Berestycki, Arch. Rat. Mech. Anal. 82 pp 313– (1983)
[5] Berestycki, C. R. Acad. Sc. 293 pp 489– (1981)
[6] Bona, Proc. R. Soc. Lond. A 344 pp 363– (1975)
[7] Cazenave, Nonlinear Analysis, Theory, Methods & Applications 7 pp 1127– (1983)
[8] Cazenave, Comm. Math. Phys. 85 pp 549– (1982)
[9] Coffman, Arch. Rat. Mech. Anal. 46 pp 81– (1972)
[10] Generalization of the Korteweg-de Vries equation, Proc. Symp. Pure Math. 23, and Ann. Math. Soc., 1973, pp. 393–307.
[11] Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York, 1969.
[12] Ginibre, J. Func. Anal. 32 pp 1– (1979)
[13] Henry, Comm. Math. Phys. 85 pp 351– (1982)
[14] Holm, Phys. Lett. 98A pp 15– (1983)
[15] On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in Studies in Appl. Math. Advanced in Mathematics Supplementary Studies 8, Academic Press, New York, 1983.
[16] Laedke, Phys. Rev. Let 52 pp 279– (1984)
[17] Laedke, J. Plasma Physics
[18] McLeod, Proc. Nat. Aca. Sci. USA 78 pp 6592– (1981)
[19] Strauss, Comm. Math. Phys. 55 pp 149– (1977)
[20] Strauss, Arch. Rat. Mech. Anal. 55 pp 86– (1974)
[21] Shatah, Commun. Math. Phys. 91 pp 313– (1983)
[22] and , Instability of nonlinear bound states, preprint.
[23] Weinstein, Comm. Math. Phys. 87 pp 567– (1983)
[24] Weinstein, Siam J. Math. Anal. 16 pp 567– (1985)
[25] Self-focusing and modulational analysis of nonlinear Schrödinger equations, Ph.D. thesis, NYU, New York, 1982.
[26] On the structure and formation of singularities in solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, in press.
[27] Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974.
[28] Zakharov, Sov. Phys. JETP 35 pp 908– (1972)
[29] Zakharov, Sov. Phys. JETP 38 (1974)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.